In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

1 aStrazzullo, Maria1 aBallarin, Francesco1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/pod-galerkin-model-order-reduction-parametrized-nonlinear-time-dependent-optimal-flow01433nas a2200145 4500008004100000245016600041210006900207260001300276520087600289100002201165700001801187700002401205700002101229856003701250 2020 eng d00aReduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences0 aReduced order methods for parametrized nonlinear and time depend bSpringer3 aWe introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.

1 aStrazzullo, Maria1 aZainib, Zakia1 aBallarin, Francesco1 aRozza, Gianluigi uhttps://arxiv.org/abs/1912.0788600512nas a2200145 4500008004100000245011100041210006900152300001600221490000700237100002200244700002400266700001600290700002100306856003900327 2018 eng d00aModel Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering0 aModel Reduction for Parametrized Optimal Control Problems in Env aB1055-B10790 v401 aStrazzullo, Maria1 aBallarin, Francesco1 aMosetti, R.1 aRozza, Gianluigi uhttps://doi.org/10.1137/17M1150591